Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#2017⋊C3) |
| unambiguous semidirect product | #68727⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#127⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C3×#338 | C2×C3 |
| Frattini | C22×C4 | D3×A4 |
| socle | C2×C3 | #392⋊C3 |
| LCS | G ▷ C3×#338 | |
| UCS | C1 ◁ C4 | |
157 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2,3,2^2,3,2^4|a^2=fh,c^2=f,d^2=fg,f^2=i,c^a=ch,d^a=dgh,e^a=e^2,c^b=d,d^b=cdg,f^b=fg,g^b=hi,h^b=gh,d^c=dfh,g^c=gi,f^d=fi,g^d=gi,h^d=hi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154161);